On structures of normal forms of complex points of small $${\mathcal {C}}^{2}$$-perturbations of real 4-manifolds embedded in a complex 3-manifold

Abstract

AbstractWe extend our previous result on the behaviour of the quadratic part of a complex points of a small $${\mathcal {C}}^{2}$$ C 2 -perturbation of a real 4-manifold embedded in a complex 3-manifold. We describe the change of the structure of the quadratic normal form of a complex point. It is an immediate consequence of a theorem clarifying how small perturbations can change the bundle of a pair of one arbitrary and one symmetric $$2\times 2$$ 2 × 2 matrix with respect to an action of a certain linear group.